Let the symmetric group $S_5$ act by permutation on the set $X:=\{S \subset \{1,2,3,4,5\}:|S|=2\}$ and denote by $V$ the associated complex $S_5$-representation, with $\chi:S_5 \rightarrow \mathbb{C}$ its character. I'm supposed to compute $\chi(\sigma)$ for $e, (2,3), (1,2,3), (1,2,3,4), (1,2,3,4,5), (1,2)(4,5), (1,5)(2,3,4)$. I know the character is the trace of the matrix that each element of $S_5$ is mapped onto, and the standard presentation of $S_5$ in terms of transpositions, however I'm young in this field, so I can't really see what the are actual matrices that the transpositions $(1,2), (2,3), (3,4), (4,5)$ are mapped to are in this case. Another difficulty is that I can't quite see how to represent the elements of $X$ as elements of a vector space. Could it be the $5$-dimensional complex vectors with only $2$ entries equal to $1$ and all other entries vanishing? Thanks in advance!
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$\begingroup$ Probably you should ask instead «what is the complex representation associated to that action on the set $X$», as that seems to be what you are misssing. $\endgroup$– Mariano Suárez-ÁlvarezCommented Feb 10, 2022 at 2:28
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$\begingroup$ @MarianoSuárez-Álvarez Yes this is exactly what I can't seem to figure out, any clues? $\endgroup$– JBuckCommented Feb 10, 2022 at 2:36
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$\begingroup$ There's nothing magic about the vector space in question. If $G$ acts on a set $\Omega$ of size $n$ we just form the vector space of formal sums $V=\{\sum_{\omega\in\Omega} \lambda_\omega \omega\mid \lambda_\omega\in\mathbb{C}\}$. And the last thing you want to know are the matrices: all you need to know is that they are permutation matrices, so that the trace of the matrix representing $g$ is determined by how often you get a $1$ on the diagonal .. so when does $\omega\cdot g=\omega$ for your various $g$? $\endgroup$– ancient mathematicianCommented Feb 10, 2022 at 15:37
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$\begingroup$ Highly relevant: math.stackexchange.com/questions/308722/… $\endgroup$– ancient mathematicianCommented Feb 10, 2022 at 15:59
1 Answer
The set $X$ has ten elements, and $S_5$ is acting by permutation on those elements. So think of a ten-dimensional vector space, with axes labeled by the elements of $X$. Every element of $S_5$ then turns into a 10-by-10 permutation matrix. The character of an element is going to be the trace of the matrix. As it turns out, this only depends on the conjugacy class of the element (since $g' = h^{-1}gh$ implies $\phi(g') = \phi(h)^{-1} \phi(g) \phi(h)$ implies $Tr(g') = Tr(g)$, where $\phi$ is the matrix representation).
If $g=e$, what is the corresponding 10-by-10 matrix $\phi(g)$? If $g = (2,3)$, what is the corresponding matrix? Try labelling rows and columns by elements of $X$ and see what happens.
Hint: Under what circumstances does the matrix have a 1 on the diagonal? Under what circumstances does it have a 0?
